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Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds

  • Anar Akhmedov
  • Sümeyra Sakallı
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Abstract

We study negative spheres obtained from exceptional sections of Gurtas Lefschetz fibrations. Starting with the known system of 4n disjoint (−1)-sections, we show that a marked double sum contains 4n symplectic (−2)-spheres and 2n smooth (−4)-spheres obtained by pairwise tubing. By carrying the same sections through a fourfold sum, we instead obtain 4n disjoint symplectic (−4)-spheres and hence symplectic rational blowdowns. We then consider the Lefschetz fibrations on knot-surgered elliptic surfaces. Their branched-cover description gives 2n branch sections together with simultaneous geometric duals arising from node resolution. In the knot-surgery double these sections glue to symplectic (−4)-spheres, while the duals make the complement of every subcollection simply connected. The resulting rational blowdowns provide simply connected exotic symplectic four-manifolds under the parity condition stated in the main theorem. We also record the boundary-multitwist relation determined by the Gurtas sections.

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Cite

BibTeXRIS
BibTeX
@misc{akhmedov2026gurtas,
  title = {{Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds}},
  author = {Anar Akhmedov and Sümeyra Sakallı},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17871},
  eprint = {2609.17871},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17871},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds
AU  - Anar Akhmedov
AU  - Sümeyra Sakallı
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17871
UR  - https://arxiv.org/abs/2609.17871
AB  - We study negative spheres obtained from exceptional sections of Gurtas Lefschetz fibrations. Starting with the known system of 4n disjoint (−1)-sections, we show that a marked double sum contains 4n symplectic (−2)-spheres and 2n smooth (−4)-spheres obtained by pairwise tubing. By carrying the same sections through a fourfold sum, we instead obtain 4n disjoint symplectic (−4)-spheres and hence symplectic rational blowdowns. We then consider the Lefschetz fibrations on knot-surgered elliptic surfaces. Their branched-cover description gives 2n branch sections together with simultaneous geometric duals arising from node resolution. In the knot-surgery double these sections glue to symplectic (−4)-spheres, while the duals make the complement of every subcollection simply connected. The resulting rational blowdowns provide simply connected exotic symplectic four-manifolds under the parity condition stated in the main theorem. We also record the boundary-multitwist relation determined by the Gurtas sections.
ER  -